The key step in the potential retrieval is to determine the right correction \(\Delta V^{(j)}(r)\) .
This correction was elegantly demonstrated to depend on the covariance of the measured pair correlation function [1]}.
Explicitly, it is convenient to introduce the function \(S(r)~=~2\pi r g(r)\)  (valid for the 2D case), and expand it in its Taylor series in \(\Delta V^{(j)}(r)\) :
\(\Delta \langle S^{(j)}\rangle (r_\alpha )= \sum _{r_\beta } \frac{\partial \langle S^{(j)} \rangle (r_\alpha )}{\partial V^{(j)}(r_\beta )}\Delta V^{(j)}(r_\beta ) + O(\Delta V^2).\) 
